I'm surprised a bit at how few people (just 1, I think) mentioned Super Mario Sunshine. And in general, people on these types of sites seem disappointed by it. It was one of the first games I and my son played when we got the Cube. We both loved it, and he still plays it about once a week 1 1/2 years later. I haven't played it since we finished it, but before that I often stayed up after him and continued to play to get through some of the more difficult parts. We never were able to get all the shines, which is quite a challenge.
I had been out of console gaming since the Colecovision (except for a brief stint with an NES - didn't like it, went to computer gaming). When I put in Rogue Leader I was blown away - THIS is the game I wanted when I was a kid (and the original Star Wars movies were coming out). Since you have an Xbox the graphics probably won't blow you away quite so much, but for me that was amazing. I would also recommend the sequel, which gets a lot of hate for some reason. Yeah, the ground missions aren't so great (particularly the Luke/Yoda one which is pure annoyance) but the rest is pretty good. And you can get both pretty cheap now. The second has a full cooperative version of the first game, which is pretty fun too.
I'm almost done with Tales of Symphonia - an excellent game, just clear your schedule for a few weeks. My family also spent a lot of time on Animal Crossing, but that petered out after about 6 months - can't wait for the DS version though.
Another game my son really likes is Wario World - it is very short (compared to Mario Sunshine) and kind of a strange game but fun if you like beating on stuff - and it is another you can get pretty cheap (definitely worth $10 or so - but being a CAG, I wouldn't pay any more).
I have Baten Kaitos and will play that after Tales.
We finished Wind Waker but I was not all that impressed - like Mario much better if forced to pick between the two. Started playing both Pikmins but haven't finished either - they are fun though, just too many games right now! Paper Mario is an excellent game as well - we did finish that one.
I haven't played Starfox Adventures (though I have it) but Starfox Assault is pretty fun - short though, and probably not worth $50. I got it as a $30 Gamerush preorder. You have plenty of other games to go through first and then you can get it in a 2 for $30 deal when it becomes PC. I have been having a tougher time getting medals in Starfox Assault then I ever did in Rogue Squadron (and in parts the games are similar).
So, if I had to put together a MUST have list (counting some of the sequels together) for the Gamecube it would be, in no particular order:
- Animal Crossing
- Super Mario Sunshine
- Star Wars: Rogue Squadron 2 (and I'll include 3 here as well)
- Metroid Prime and Echoes (though I haven't gotten around to playing Echoes)
- Mario Kart
- Tales of Symphonia
- Pikmin 1/2
Note that I have less than zero interest in that whole Resident Evil thing and in FPS's, so my lists will never include those...
Other fun games that maybe aren't must haves: Mario Tennis, Starfox Assault, SSBM (meh, its OK, but I waited 1 year before getting it - note that besides ones with my son I don't do any multiplayer games - the following one excepted since my daughter will play also), Mario Party (any number - get what's cheapest - we have 4 and 6), Luigi's Mansion (again, one we played right away and liked - get cheap though), Wind Waker (sorry this is where I put it), Wario World, Viewtiful Joe (have played through 1/2 of #1, none of #2 - thanks to Kmart got both for less than $20 combined!), Mario Golf (I'm not into golf at all and not much into sports games but this is pretty fun along with the GBA version).
I'm sure there are others - of course I have left off the multi-platform titles that I have played like Burnout 2, etc, since you have an Xbox.